In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. In words, the algorithm is described as follows: For the N-body problem the result is: with The vector is the center of mass of all the bodies and is the relative coordinate between the particles 1 and 2: In the calculations can be useful the following identity .
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| - Coordenades de Jacobi (ca)
- Jacobi-Koordinaten (de)
- Coordenadas de Jacobi (es)
- Jacobi coordinates (en)
- 雅可比坐標 (zh)
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| - En teoria de sistemes de moltes partícules, les coordenades de Jacobi s'utilitzen per simplificar la formulació matemàtica. Són comunament utilitzades en l'estudi de molècules poliatòmiques i reaccions químiques, i en mecànica celeste. Les coordenades de Jacobi es defineixen seguint el següent algorisme: Siguin mj i mk les masses de dos cossos que són substituïdes per un cos nou de massa virtual M = mj + mk. Les coordenades de posició xj i xk són substituïdes per les seves posicions relatives rjk = xj-xk i pel vector posició del seu centre de masses Rjk = (mj xj + mk xk)/(mj + mk). (ca)
- Die Jacobi-Koordinaten sind ein System verallgemeinerter Koordinaten für n-Körpersysteme in der Physik. Sie werden insbesondere in der Himmelsmechanik und der Betrachtung mehratomiger Moleküle und chemischer Reaktionen verwendet. (de)
- En la teoría de sistemas de muchas partículas, las coordenadas de Jacobi se usan con frecuencia para simplificar las fórmulas matemáticas. Estas coordenadas son especialmente comunes en el tratamiento de moléculas poliatómicas y reacciones químicas, y en mecánica celeste. (es)
- 在多体系统的研究中，常用雅可比坐標来简化数学计算。这一坐标系统可以用于多个领域，尤其是天体物理，以及多原子分子和化学反应 。一个用于N体问题建立雅可比坐标的算法是利用二叉树。这一算法可以这样描述： 质量分别为mj和mk的两个物体用一个质量为M = mj + mk的虚拟物体代替。同时，用相对坐标向量rjk = xj − xk和质心坐标向量Rjk = (mj qj + mkqk)/(mj + mk)来替代两个物体原来的坐标向量xj和xk。二叉树中的一个节点即为这一虚拟物体。它有两个子节点，左子节点为mk，右子节点为mj。对N-1个物体重复以上步骤。 四体问题的结果是： 其中： 向量R是所有物体的质心： (zh)
- In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. In words, the algorithm is described as follows: For the N-body problem the result is: with The vector is the center of mass of all the bodies and is the relative coordinate between the particles 1 and 2: In the calculations can be useful the following identity . (en)
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| - En teoria de sistemes de moltes partícules, les coordenades de Jacobi s'utilitzen per simplificar la formulació matemàtica. Són comunament utilitzades en l'estudi de molècules poliatòmiques i reaccions químiques, i en mecànica celeste. Les coordenades de Jacobi es defineixen seguint el següent algorisme: Siguin mj i mk les masses de dos cossos que són substituïdes per un cos nou de massa virtual M = mj + mk. Les coordenades de posició xj i xk són substituïdes per les seves posicions relatives rjk = xj-xk i pel vector posició del seu centre de masses Rjk = (mj xj + mk xk)/(mj + mk). (ca)
- Die Jacobi-Koordinaten sind ein System verallgemeinerter Koordinaten für n-Körpersysteme in der Physik. Sie werden insbesondere in der Himmelsmechanik und der Betrachtung mehratomiger Moleküle und chemischer Reaktionen verwendet. (de)
- En la teoría de sistemas de muchas partículas, las coordenadas de Jacobi se usan con frecuencia para simplificar las fórmulas matemáticas. Estas coordenadas son especialmente comunes en el tratamiento de moléculas poliatómicas y reacciones químicas, y en mecánica celeste. (es)
- In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. In words, the algorithm is described as follows: Let mj and mk be the masses of two bodies that are replaced by a new body of virtual mass M = mj + mk. The position coordinates xj and xk are replaced by their relative position rjk = xj − xk and by the vector to their center of mass Rjk = (mj qj + mkqk)/(mj + mk). The node in the binary tree corresponding to the virtual body has mj as its right child and mk as its left child. The order of children indicates the relative coordinate points from xk to xj. Repeat the above step for N − 1 bodies, that is, the N − 2 original bodies plus the new virtual body. For the N-body problem the result is: with The vector is the center of mass of all the bodies and is the relative coordinate between the particles 1 and 2: The result one is left with is thus a system of N-1 translationally invariant coordinates and a center of mass coordinate , from iteratively reducing two-body systems within the many-body system. This change of coordinates has associated Jacobian equal to . If one is interested in evaluating a free energy operator in these coordinates, one obtains In the calculations can be useful the following identity . (en)
- 在多体系统的研究中，常用雅可比坐標来简化数学计算。这一坐标系统可以用于多个领域，尤其是天体物理，以及多原子分子和化学反应 。一个用于N体问题建立雅可比坐标的算法是利用二叉树。这一算法可以这样描述： 质量分别为mj和mk的两个物体用一个质量为M = mj + mk的虚拟物体代替。同时，用相对坐标向量rjk = xj − xk和质心坐标向量Rjk = (mj qj + mkqk)/(mj + mk)来替代两个物体原来的坐标向量xj和xk。二叉树中的一个节点即为这一虚拟物体。它有两个子节点，左子节点为mk，右子节点为mj。对N-1个物体重复以上步骤。 四体问题的结果是： 其中： 向量R是所有物体的质心： (zh)
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